Godeaux-Serre Varieties with Prescribed Arithmetic Fundamental Group

TL;DR

This paper demonstrates that for any field and dimension, one can construct smooth projective varieties whose arithmetic fundamental groups realize any specified extension of the absolute Galois group by a finite group.

Contribution

It establishes the existence of geometrically connected smooth projective varieties with prescribed arithmetic fundamental groups for any field and dimension.

Findings

Any extension of the absolute Galois group by a finite group can be realized as an arithmetic fundamental group.

Construction applies to all fields and dimensions greater than or equal to 2.

Provides a method to realize specific Galois extensions geometrically.

Abstract

We show that for any given field $k$ and natural number $r\geq2$, every continuous extension of the absolute Galois group $\mathrm{Gal}_k$ by a finite group is the arithmetic fundamental group of a geometrically connected smooth projective variety over $k$ of dimension $r$.